Much on your list is easily preventable by driving sensibly, paying attention without being afraid. Confidence driving comes with going out and getting mileage in your vehicle.
You are very UNLIKELY to tip over unless you drive up on something higher on one side, lIke a fallen log or deep hole. Driving on familiar roads and trails reduces the chance of “surprises” that could tip you. Control your speed, especially during turns, corners, circles. Have to say bigger wheels on carts help “bridge” ruts and holes better than smaller wheels. Carts (2 wheels) are usually wider vehicles than 4 wheeler, which adds to stability to prevent tipping. Carts move with the animal, forward and back, turn with him, the outfit being a straight line. This is why carts are highly recommended for starting new horses, new drivers.
Not driving thru brush, untrimmed woods, skinny gates, should prevent 99% of any chance of snagging or getting tangled on something. Do not drive tight to fences where you can hook the shaft or hub on something. Give yourself plenty of room. Look ahead, slow down if something ahead is questionable. Stop, look it over, have a passenger walk it if possible, checking for hidden things, then continue if it appears safe. No shame in backing up, turning around at questionable places! Carriages and ridden horses need different eyes looking to decide what is safe to go forward into.
4 wheeled vehicles are hinged for easier turning, more comfortable ride, but the hinge allows vehicle to fold or jack-knife while backing and on sharp turns. ESPECIALLY at speed. So they are for a more experienced Driver and horse. 4 wheels is a logical step up because they have more skills, miles driving, when they change from a 2 wheel cart…
Equipment failure is less likely to happen if you pay attention to the harness, check your vehicle regularly. With the advent of synthetic harness, people do not take it apart regularly to clean and condition, as we did with our leather harnesses. This lets you run your hands over the straps, feel for wear, broken stitches, enlarging holes, thin leather, as you wipe thinge down, reassemble the harness. Taking reins off with buckled billets instead of snaps, has you feeling, hopefully LOOKING, at that piece of leather each drive to see and feel the wear before a rein fails you. Vehicles should be gone over at least yearly in Spring to check tightness of bolts, nuts, spring shackles, shafts, disassembly of wheels to degrease, check bearings for wear, put in new grease and reseal the wheel. If you cross water often, your wheels will need greasing more often in case water gets in there. If your vehicle has brakes you will need to drain and refill the fluid because it collects moisture from the air. Fluid should be clear, any milky coffee color, cloudiness, means you have water in the brakes. Not changing fluid can eat up brake seals, so brakes will fail later.
If you don’t feel capable doing vehicle upkeep, have a professional person do it for you. There is a recent book sold by Driving Digest, the collection of articles they published written by Dave Quist. It is very thorough in explaining vehicle maintenance, how to do things yourself and why. Well worth ihe purchase price.
For horses falling? That seems to be “an act of God.” Not always preventable, but pretty rare if you are going at modest speeds, straighter lines of travel like recreational drives. This is where your “whoa training”, your horse trusting you, comes in. Keep firm hold on the reins, don’t give slack even if he does stay still. Get up, get out of vehicle, look at how things are. Talk to him calmly, keep reins tight as you get closer. Is he on one side? Belly? On a single, he is attached at traces, shafts, breeching holdback straps. Modern synthetic harness often has quick releases that make getting undone faster, simpler. You can unsnap the traces, backstrap, shaft loops, then let him arise and move forward. Breeching falls off with backstrap released.
Each situation is different, though maintaining control of his body is most important. It often seems driving horses are more accepting of restraint, may not fight when downed. However at the LEAST loosening of harness or rein tension they will try to rise. Having a second person if you have one, sit on his throatlatch keeping the head down while you work on horse, is very helpful in keeping him down and quiet. Stay out from between hooves and belly area to prevent YOU getting injured in getting hIm freed. There is no “plan of action” in steps to follow for accidents beyond keeping horse quiet while you assess and deal with things. Harness cutting in accidents is VERY seldom needed, plus horse will try to get up with release of harness tension. NEVER cut reins, you lose all control of horse and events following.
Think more about the good stuff, enjoy going out in the carriage with your horse, how fun drIvins is!. Improve your driving skills, whip handling to hit a loose dog,. Appear confident, practice makes it true! Drive sensibly, worry about trouble when it happens. Otherwise worrying will creep down the reins, affect the horse, ruin the day.